Tapered spiral welded structure

ABSTRACT

A method for creating a tapered spiral welded conical structure where the overall shape of the cone is first graphically slit axially and unwrapped, and then a series of construction arcs and lines are created to form the edge lines of a strip that can then be wrapped (rolled) to form a tapered conical structure. The edges of the spirally wound strip can be welded together, and a very large conical structure can thus be achieved. Various construction options are presented from a constant width strip to strip made from straight segments. Equations are given for the formation of the strips to enable those skilled in the art of spiral welded tubing to practice the invention.

This invention relates to a method for creating tapered spiral weldedstructures from strips of material that either have been precut orformed and then are rolled and welded to form a tapered spiral weldedstructure, or a straight strip of material is notched and weldedtransversely in addition to being spirally welded. Such tapered towerscan be manufactured on-site to enable very large diameter structuraltowers, not capable of being trucked to site.

STATEMENT FEDERALLY FUNDED RESEARCH

No federal funds were used in the development of this invention.

FIELD OF THE INVENTION

The present application relates to a method for defining the shape of astrip of metal that can then be formed into a spirally welded taperedconical structure.

BACKGROUND OF THE INVENTION

A spiral welded conical structure that could be made on the constructionsite could be far more efficient than current designs of supportstructures such as wind turbine towers. It could result in lighterweight for the same or greater height, compared to traditional towersthat are transported to site and bolted together. Existing spiralwelding technology is incapable of producing the required conicalstructure without significant deformation of the metal stock, a processthat is impractical to achieve on site.

U.S. Pat. No. 2,038,576 describes forming a cone from a piece of papercut into an arc segment; however, the method does not allow for multiplewraps as would be required to form a large diameter cone from a singlenarrow strip of material; nor does the method have any logical extensionto evolve into such a required shape as the radial edges of the paperforce an axial seam and thus are inherently limiting on the length ofthe cone that can be formed to be less than the radial length of theedges.

U.S. Pat. No. 2,008,423 describes making a hollow tapered shaft, such asfor a golf club, from a uniform or tapered width strip of material. Theinventor properly notes that as the strip is wound to form the taperedcone, the helix angle changes which causes the edges to spread apart,and that this is prevented by winding the strip onto a mandrel whileunder tension which allows the strip to be deformed so the pitch canchange without causing buckling of the strip. While suitable for forminga shaft such as golf club, it would not be possible to provide a mandrelfor a very large structure, such as a wind turbine tower.

U.S. Pat. No. 3,997,097 describes forming a tapered tube by firstfeeding a strip between forming rollers that taper the material acrossits width as a slight curvature is imparted to the strip to enable thehelix angle to change as the tapered tube is formed. This patentdescribes “known prior art” but not U.S. Pat. No. 2,008,423, whichdescribes the critical function of tensioning a constant width stripwhile forming it over a mandrel although the deformation described issimilar. While a mandrel is not required for this case, the order ofsize of the machine required to form the strip in this manner would makeit difficult to realize for very large cones such as that required forlarge structures such as a windmill tower. Feeding a previouslymanufactured bent strip to the machine is mentioned, but no algorithmfor what the shape needs to be is mentioned.

U.S. Pat. No. 4,082,211 (by the same inventor as U.S. Pat. No. 3,997,097and using similar figures) describes a process for producingfrusto-conical tubes of gradually varying diameter by winding stripstock into a chain of frusto-conical helical turns and joining the edgesas they are wound by automatically continuously sensing the variation intube diameter as it is being produced, automatically governing saidvariation responsive to rotation of a, pair of stock-edge squeezingrollers, and automatically feeding back sensed information of saidvariation in tube diameter to effect the desired adjustment in spacingbetween said rollers, and for automatically controlling the rate atwhich said variation takes place. This will create the in-planecurvature required, but it is not applicable to the large 2 m widestrips needed for large structures, as the rolling forces in the coldstate would be titanic and it would not be practical to heat the strip.Furthermore, the described feedback system is only capable of findingthe required curvature of the strip during the cone forming process.

U.S. Pat. No. 6,339,945 describes a spiral tube forming system forforming a strip into a spiral tube; a strip in-feed system adapted forfeeding a strip to the pipe forming system; and computer-controlledmeans for continuously varying the angular orientation of the tubeforming system relative to the strip in-feed system to selectively varythe diameter of the forming tube. It uses a metal strip of constantwidth and continuous change in feeding angle to vary diameter. There isno mention of the required geometry to form a cone. There is no mentionof cutting the metal strip. The apparatus disclosed does not have theability to bend the strip in-plane, and even it was to do so, the stripwould buckle. To form the strip in-plane curvature, the strip must bevery hot, and hence is best form at the mill when it is made. If thestrip were provided from the mill with the desired curvature, feedingangle control is necessary as described in U.S. Pat. Nos. 6,732,906,3,997,097, and 1,914,976.

U.S. Pat. No. 6,732,906 describes a method for meeting the varying helixangle requirement by imparting out-of-plane waviness to one edge of anotherwise constant width strip to impart an overall slight in-planecurvature needed to form the varying helix angle as 4 tapered cone isformed. However, this could decrease the buckling strength of the tower.Although it uses a metal strip of constant width which is bent in planeusing corrugating rolls there is no mention of the required geometry toform a cone, which as shown by the present invention, is a complexnon-obvious shape. There is no mention of cutting the metal strip.

For large wind turbine towers, for example, it will be desirable to formtapered (conical) towers on-site so the base can be very large indiameter. Prior art to this effect includes U.S. Pat. No. 3,030,488where a metal strip spool rests on its own weight on its drivingassembly. The method describes sensing the strip edge position beforethe weld and creating a feedback loop including means to vary thefeed-in angle. A stated goal of this method is to decrease the number ofelements and thus provide a “transportable machine by which tubes can bewelded at the point of use” [1-29].

The need for taller towers with larger diameter bases is discussed belowin the context of FIGS. 9 and 10. One way that industry has currentlybeen trying to meet this need is through segmented designs. Currently,cylindrical segments are used, but to get to large diameters, thecircular segments can be further segmented into arcs. As an example, seeUS patent application “Method of constructing large towers for windturbines (2003, application Ser. No. 10/549,807): “In order to transportlarge size windmill towers, the invention suggests a steel tower (1) fora windmill, comprising a number of cylindrical or tapered tower sections(2), at least the wider sections (2) of which being subdivided into twoor more elongated shell segments (3), which combine into a completetower section (2) by means of vertical flanges (6) tightened together,e.g., by bolts (10), said shells being also provided with upper andlower horizontal flanges (4), respectively; to allow interconnection oftower sections (2) one on top of the other.” [Abstract]. However, thisresults in great complexity and an enormous number of bolts to beinstalled and then periodically checked.

OBJECTS OF THE INVENTION

An object of this invention, therefore, is to provide a geometricconstruction method for forming a strip of material such that it canthen be wrapped onto the surface of an imaginary tapered conical formand be spirally welded to form a tapered conical structure.

A further object of the invention is to provide analytical expressionsfor forming a strip such that it can be wrapped to form a spiral weldedtapered conical structure.

A further object of the invention is to provide analytical expressionsfor forming a strip such that it can be wrapped with as many revolutionsas desired to form a spiral welded tapered conical structure, such aswould be required to make a large diameter conical structure from arelatively narrow strip of material.

A further object of the invention is to provide analytical expressionsfor forming a strip such that it can be wrapped to form a conicalstructure that can be edge-butt welded along the length of the spiral.

And yet another object of the invention is to allow a continuous stripas defined above to be made and then rolled up into a coil and thenunwound from the coil and then formed into the desired long taperedcone.

Other and further objects will be explained hereinafter and moreparticularly delineated in the appended claims.

SUMMARY

A method is presented for forming a tapered spiral welded conicalstructure where the overall shape of the cone is first graphically slitaxially and unwrapped, and then a series of construction arcs and linesare created to form the edge lines of a strip that can then be wrapped(rolled) to form a tapered conical structure. The edges can be spiralwelded together, and a very large structure can be achieved. Variousconstruction options are presented from a constant width strip to stripmade from straight segments. As and example, a constant width strip usedto form a typical tower (2 m wide strip, 5.5 m to 2.5 m diameter cone,100 m high) would have an in-plane radius of curvature that varies fromaround 180 m down to 80 m. Equations are given for the formation of thestrips to enable those skilled in the art of spiral welded tubing topractice the invention.

DRAWINGS

The present invention can best be understood in conjunction with theaccompanying drawing, in which:

FIG. 1 shows a wind turbine atop of a tapered spiral welded cone;

FIG. 2a shows a cone with one wrap of a sheet to make the cone;

FIG. 2b shows a cone with multiple wraps of a sheet to make the cone;

FIGS. 3a-3d show the steps required to create a strip of material thatcan wrap in a spiral to form a cone with 3 wraps with the edges of thestrip abutting each other so they can be welded together;

FIGS. 4a-4c show different types of strip shapes that can be formedusing the generalized method of this invention;

FIG. 5a shows how multiple strips can be formed so a multi-start wrapcan be created,

FIG. 5b shows the strip from FIG. 3d that can be wrapped to form a conewith 4 wraps

FIG. 5c shows how the strip to be wrapped to form a cone can also besegmented;

FIG. 6 shows a constant width straight-sides strip (as depicted in FIG.4a ) being constructed where the segments are straight segments withtriangular regions cut from between them;

FIG. 7 shows a constant width arc-shape sides strip being constructedwhere no segments need to be cut from between regions, but the radii ofcurvature varies;

FIG. 8 shows an involute spiral shape strip that can be formed to wrapinto a conical structure;

FIG. 9 shows a logarithmic spiral shape strip that can be formed to wrapinto a conical structure;

FIG. 10 shows a graph of the ratio of estimated tower mass versus basediameter (m) for a standard 80 m wind turbine tower designed to supporta 1.5 MW turbine;

FIG. 11 shows a graph of the ratio of estimated tower mass versus basediameter (m) for a standard 100 m wind turbine tower designed to supporta 2.5 MW turbine,

PREFERRED EMBODIMENT(S) OF THE INVENTION

FIG. 1 shows a wind turbine 11 atop a tapered tower 10. Such towers aretypically manufactured in segments from axially welded steel plate thatare transported to site and assembled. This necessitates large boltedflange connections every 20 meters or so and the largest diameter thatcan be transported is typically 4-4.3 meters. By forming the sectionson-site, diameters of 5 meters or more could be formed. Since thebending strength goes with the cube of the diameter, even a 10% increasein diameter represents a 33% increase in strength for only a 10%increase in weight if the same wall thickness structure is used. Goingfrom a 4 meter diameter structure to a 5 meter diameter structureincreases the strength by almost 100%. Hence there is a clear need to beable to make large conical towers on site so they can be used to supportmachines such as wind turbines shown in FIG. 1.

As discussed above, however, there has not been a simple and effectiveway to accomplish manufacturing of a tapered tower (conical structure)that does not require in-plane deformation of the material by themachine. Machines that could form the large wide strips that would beneeded to form a large tower on site would not be practical, unlessperhaps installed at a steel mill to operate as the hot wide strip ismade, or even economically transportable to a site for creating largetowers with base diameters on the order of 5 meters.

In the simplest form, FIG. 2a shows a cone 101 of height h, top diameterDt and base diameter Db. This conical structure is formed by a singlearc segment sheet 100 with a longitudinal joint 99. However, to form theconical section, the sheet must be rolled along its entire length whichis not practical for a very large cone. Hence as shown in FIG. 2B aspiral wrap section 103 with spiral joint 102 is preferred to form anequivalent cone 104. The challenge is that the helix angle changes withheight if a constant width strip is used, and this then necessitateschanging radii of curvature of the strip. The present invention providesa generalized method for determining the geometry of any and all stripsthat can be wrapped onto the surface of the desired cone withoutrequiring in-plane deformation during the wrapping process.

FIGS. 3a-3d show how the present invention provides a robust method fordefining the geometry of a flat strip 35 equivalent to annulus segment100 that can be rolled to form a conical structure 101. Starting withthe desired cone geometry 101, split and unwrap the cone to form a flatwedge-shaped segment 100 of an annulus with a large outside diameteredge (radius R_(o)) and a smaller inside diameter edge (radius R_(i)),where the larger diameter edge arc length equals the circumference ofthe cone base and the smaller diameter edge arc length equals thecircumference of the cone top, and straight sides connecting the edgeswith an angle α between them; hence the parameters R_(o), R_(i), and αare defined as:

$R_{o} = \frac{D_{b}\sqrt{{4h^{2}} + \left( {D_{b} - D_{t}} \right)^{2}}}{2\left( {D_{b} - D_{t}} \right)}$$R_{t} = \frac{D_{t}\sqrt{{4h^{2}} + \left( {D_{b} - D_{t}} \right)^{2}}}{2\left( {D_{b} - D_{t}} \right)}$$\alpha = \frac{\pi \; D_{b}}{R_{o}}$

Where D_(t) is the top diameter of the desired cone, D_(b) is the basediameter, and h is the height of the desired cone.

Draw a construction curve 31, which can be continuous or segmented asmay be desired, from a vertex 30 a on the edge 28 on the outside arc 27of radius R_(o) inward to intersect a construction circle 29 whoseradius is equal to and concentric with the inside arc 26 of radiusR^(i). The angle β between the radial rays passing from the center ofthe annulus 30 c through the start and stop points of the constructioncurve 31 is equal to the product of the number of desired wraps on thecone and the angle α between the straight sides 28 and 25 of the annulussegment. In FIG. 3b , the angle β is shown to be 3α, and hence thesegment 35 in FIG. 3d would make 3 wraps to form a cone. Partial wrapsare acceptable.

FIG. 3c shows the next step in the method: copy the construction curve31 and rotate the copy about the center 30 c of the annulus segment byan angle α equal to the angle α between the straight sides 28 and 25 ofthe flat wedge-shaped segment; and connect the vertices of the originalconstruction curve 31 and the copied curve 32 with arc segments 33 and27 concentric and with equal radius to the inner and outer annuli radiirespectively to form a closed section 35 which can then be wrapped toform the original cone formed by the unwrapped single segment 100.

When forming a cone, the strip must be fed into the coiling system atthe correct in-feed angle ϕ, also known as the helix angle. For allstrips other than those formed by a logarithmic spiral, shown in FIG. 9,the in-feed angle must vary as the cone is formed. The required in-feedangle can be found on the construction by drawing a radial line 301 fromthe point 30 c at the center of the annulus to a point 302 on theconstruction curve. The angle ϕ between the line 303 tangent to theconstruction curve at this point 302 and radial line 301 is the helixangle of the cone when this section of the strip enters the cone.

Every strip that can be formed into a cone without overlap or in-planedeformation can be found using this construction technique. Thegeometric properties of the strip are defined by the choice inconstruction curve 31. There are a number of special cases that may makeconstruction of a strip coiling machine easier.

For example, for most strips the strip in-feed angle must vary duringcoiling to form the cone. The exception to this is the strip formed froma logarithmic spiral 48 shown in FIG. 9. In polar coordinates this curveis defined as:

r(θ) = e^(ϑ cot (φ))$r = \left. \frac{D_{t}\pi}{\alpha}\rightarrow\frac{D_{b}\pi}{\alpha} \right.$

Where r(θ) is the distance from the center of the annulus 30 c to pointson the logarithmic spiral construction curve, and θ is the polar anglein radians to these same points measured from an arbitrary zeroreference angle. The start and stop points of this curve are defined bythe dimensions of the desired cone as defined by D_(t), D_(b) and α.This curve, and a copy of it rotated about the origin by α, form theedges of a curved strip that can be fed into a cone at a constant helixangle ϕ. The strip will not have a constant width as shown in FIG. 4 b.

Flaying a non-constant width strip may make material handling moredifficult for the cone forming machine, and may result in more wastageof raw materials. There are a number of solutions that result in aconstant width strip,

One constant width strip solution can be formed from a series of annularsegments as shown in FIG. 7. Each of these annulus segments has anincluded angle α equal to the included angle of the originalconstruction annulus segment 30. The difference between the inner radiusr^(i,n) and outer radius r_(o,n) is held fixed allowing these annulussegments to be joined into a long strip with constant width S_(w). Theinner and outer radii of the n^(th) segment can be calculated based onthe cone dimensions D_(t), D_(b) and α the desired strip width S_(w):

$r_{i,n} = {\sqrt{\left( \frac{\pi \; D_{b}}{\alpha} \right)^{2} - \left( \frac{S_{w}}{2\mspace{14mu} \tan \mspace{14mu} \left( {\alpha \text{/}2} \right)} \right)^{2}} - {S_{w}\left( {{1\text{/}2} + n} \right)}}$r_(o, n) = r_(i, n) + S_(w)

The initial partial annulus segment 60 a is found by setting n to zero,and using this equation to determine the size and shape of the annulussegment. The section of the strip outside of the outside radiusconstruction arc 27 must be removed so that the cone will have a flatbase. The next annulus segment 60 b is then found by setting n to oneand appending this segment to the initial segment. Further segments 60c, 60 d, 60 e . . . are found by incrementing n by one, until the top ofthe cone is reached. This results in a constant width strip that can becoiled into the desired cone. The strip will have continuous and smoothedges, but will have step changes in the radius of curvature.

Curved strip metal stock is generally not readily available so solutionsthat make use of straight strips may be preferred. One solution is astepped spiral as shown in FIG. 4a . One way to form this shape from astraight strip is to periodically removing triangular sections ofmaterial, and rejoin the edges to form the stepped spiral. FIG. 6 showsthis strip before the segments are rejoined.

In this solution, the triangles removed are all identical isoscelestriangles with a height equal to S_(w), and with the angle between theequal sides equal to α, the same angle as the included angle in theoriginal construction annulus segment 30. The distance between then^(th) and n^(th)+1 cutout is given by:

$L_{n} = {\sqrt{\left( {\pi \; D_{b}\frac{\sin \mspace{14mu} \left( {\alpha \text{/}2} \right)}{\alpha \text{/}2}} \right)^{2} - S_{w}^{2}} - {{S_{w}\left( {1 + {2n}} \right)}\mspace{14mu} \tan \mspace{14mu} \left( {\alpha \text{/}2} \right)}}$

Using this geometry, standard strip stock could be used by a spiralwelding apparatus to manufacture tapered structures. In addition to thecomponents that comprise a classic cylinder forming machine, saidapparatus must also contain a unit capable of removing triangularsegments of material, and rejoining the edges, in addition to unitscapable of continuously changing the feed angle throughout the formingprocess.

These are just a few examples of the strip geometry that can be used toform a cone. Each has advantages and disadvantages which impact thedesign of the coiling machine. The stepped spiral has the advantage ofusing industry standard hot rolled coil stock, but the additionalcross-welding step adds to machine cost and complexity and potentiallyslows the rate at which cones can be formed. If custom strip shapes areformed at the steel mill, then the involute or logarithmic spirals arelikely preferable. Selecting between these depends on the difficulty ofbuilding a coiling machine with either active in-feed angle control, orvariable width strip handling.

FIGS. 4a-4c and FIGS. 5a-5c give further examples of possible stripgeometries from a linear tapered strip 42, to a constant curvaturetapered strips 46, to triangular segments 47 a-47 c. Selection of aspecific geometry should be guided by the properties of the availablematerials, and the details of the coiling machine, which vary dependingon the cone scale and application.

In some applications, it may be desirable to simultaneously coilmultiple strips together to form the cone. This could increase the rateof cone production by allowing multiple welders to run in parallel, andalso allows for steeper helix angles for a given strip width. FIG. 5ashows how M multiple strips 45 a, 45 b, and 45 c can be formed by usingthe previously describe construction technique, but rotating theconstruction curve 31 by α/M rather than α. M such strips together thenwrap to form the cone. In a machine to make conical structures, Mnarrower strips would thus be used instead of a single strip M times aswide.

Some manufacturing techniques may have trouble forming a strip with stepchanges in the radius of curvature, so a continuous solution may bedesirable. FIG. 8 shows how this shape 47 can be found by taking theprevious annulus segment solution, and calculating the curve thatresults from making the included angle of the segments approach zero.The resulting curve is the involute to a circle centered on the origin30 c with radius equal to S_(w)/α where α is the included angle of theannulus segment 30 measured in radians. In polar coordinates, this curveis given by:

${r(x)} = {\frac{S_{w}}{\alpha}\sqrt{1 + x^{2}}}$θ(x) = tan⁻¹(x) − x from$x = \sqrt{\left( \frac{\pi \; D_{t}}{S_{w}} \right)^{2} - 1}$ to$x = \sqrt{\left( \frac{\pi \; D_{b}}{S_{w}} \right)^{2} - 1}$

Where D_(t), D_(b), α, and S_(w) are the previously defined geometricproperties of the cone and strip. The edge of the strip is defined bythe construction curve in polar coordinates r, and θ (measured inradians). The parameter x determines the distance along the involutespiral. The given bounds on x ensure that the cone starts and stops atthe correct diameter. If either of the bounds on x become imaginary,then the given strip width is too large to form the desired cone withoutcutting or overlap.

Using conical shells with large base diameters as wind turbine towers,as is made possible by the present invention, presents clear economicgains. The results yielded by a simple structural model are exposed asan illustration of this advantage as shown in FIGS. 10 and 11. Forspecified loads and tower heights from standard wind turbines, theminimum amount of steel is found so that the lowest limit, eithermaterial allowable stress or buckling stress, is not reached. Thecorresponding structure mass is readily obtained. FIGS. 10 and 11illustrate how, according to this model, for two different cases, themass of a wind turbine tower varies with its base diameter.

The reference (100%) is set for a base diameter of 4.3 m since it is thecurrent maximum base diameter for standard towers due to shippinglimitations. This curves show that the mass decreases significantly asthe base diameter is increased until a certain point. In this region,the tower is designed with a wall thickness large enough for the localstress not to reach the steel maximum allowable stress limit. As thediameter increases, the loads are shared over a larger surface area andthe stress decreases. Thus, the walls can be made thinner and theoverall mass decreases. If the base diameter is larger than a certainvalue however, the design driving mechanism changes. The wall thicknessreaches a lower limit such that the buckling stress is the lowest stresslimit. Thus, the wall thickness cannot be decreased as much anymore anda larger base leads to an increased total mass unless bucklingstiffeners are used.

This shows there is an optimal base diameter, unless buckling stiffenersare used, which depends on each specific case. However, for this type ofutility scale multi-megawatt wind turbine application, this optimum ismuch larger than the current 4.3 m. FIG. 11 indicate that a decrease ofabout 40% in the quantity of steel needed, and hence cost, can beachieved for some common utility scale wind turbines by using largediameter towers. Moreover, this potential gain can be even larger fortaller towers or larger wind turbines.

In addition, it could even be an enabling factor in making taller towersthan would otherwise not be feasible. Taller towers could turn oncemarginal sites into high value wind sites and could allow other windregimes to be reached.

This analysis shows that overcoming transportation limitations andenabling a way to use large base diameter towers is a fundamental needof the wind turbine industry. Current tower manufacture techniquesrequire heavy equipment and fixturing which cannot be made easilytransportable. Spiral welding manufacture is better suited for on-sitetower manufacture. Current spiral welding machines are incapable ofproducing the tapered towers required by the wind industry because ofthe required in-plane curvature. The current invention provides thismuch needed solution.

Manufacturing a cone from the strips described above may be easier ifthe strip overlaps itself rather than being joined edge to edge. Thisalso introduces buckling resistance if the helix angle is sufficientlysteep as is the case in the multiple strip configurations. If overlap isdesired, the strip width must be increased by the desired amount ofoverlap. Allowing the overlap to vary along the length of the cone maysimplify the required strip geometry.

Further modifications of the invention will also occur to personsskilled in the art, and all such are deemed to fall within the spiritand scope of the invention as defined by the appended claims.

1-16. (canceled)
 17. A stock material comprising: a strip of planarmaterial, the strip of planar material having a plurality of firststraight edges and a plurality of second straight edges, each firststraight edge defining a respective section of a first stepped spiral,and each second straight edge defining a respective section of a secondstepped spiral, wherein each first straight edge of the first steppedspiral is parallel to a respective second straight edge of the secondstepped spiral, wherein the plurality of first straight edges and theplurality of second straight edges are collectively dimensioned suchthat the strip of planar material is formable into a tapered structurein which the first stepped spiral adjoins the second stepped spiral. 18.The stock material of claim 17, wherein at least two of the firststraight edges have different lengths.
 19. The stock material of claim17, wherein the strip of planar material is a weldable material.
 20. Astock material comprising: a plurality of planar sections, each planarsection having a first edge and a second edge different from the firstedge, wherein the plurality of planar sections is dimensioned such that,in a first planar configuration, the first edges of the plurality ofplanar sections are spaced apart from and aligned with one another, eachsecond edge of each planar section is in a point contact with therespective second edge of another planar section, and each pair ofplanar sections in point contact with one another defines a respectivetriangular space therebetween.
 21. The stock material of claim 20,wherein each triangular space has the same dimensions as each of theother triangular spaces.
 22. The stock material of claim 21, whereineach triangular space is an isosceles triangle.
 23. The stock materialof claim 20, wherein each point contact between adjacent planar sectionsdefines a first vertex of the respective triangular space between theadjacent planar sections.
 24. The stock material of claim 20, wherein asecond vertex and a third vertex of each respective triangular space arealigned with the first edges of the plurality of planar sections. 25.The stock material of claim 24, wherein the plurality of planar sectionsare movable, relative to one another, from the first planarconfiguration to a second planar configuration in which each triangularspace is collapsed and the plurality of planar sections collectivelyform a stepped spiral.
 26. The stock material of claim 20, wherein eachplanar section of each pair of adjacent planar sections is adjoined toone another at the point contact therebetween.
 27. The stock material ofclaim 20, wherein at least one planar section of the plurality of planarsections is an isosceles trapezoid.
 28. The stock material of claim 20,wherein the first edges of the plurality of planar sections are linear.29. The stock material of claim 20, wherein at least one of the secondedges is linear.
 30. The stock material of claim 20, wherein at leastone of the first edges is parallel to at least one of the second edges.31. The stock material of claim 20, wherein the planar sections of theplurality of planar sections are formed of a weldable material.
 32. Thestock material of claim 20, wherein the first edges of at least two ofthe plurality of planar sections have different lengths.
 33. The stockmaterial of claim 20, wherein a respective maximum width is the same foreach planar section in a direction perpendicular to the first edges ofthe planar sections.